Congruency and Similarity of Triangles
What is Congruency of Triangles?
Congruency means harmony or compatibility.
Two triangles are said to be congruent, if they are equal in both shape and size. That is, they are identical in every way - one of them can be made to superimpose on the other.
It is denoted by the symbol ≅.
For example, if ∆ABC ≅ PQR, then:
Corresponding angles are congruent: ∠A = ∠P; ∠B = ∠Q; ∠C = ∠R
Corresponding sides are congruent: AB = PQ; BC = QR; AC = PR
Properties related to Congruency of triangles
Property 1
If two triangles are congruent, then all of their corresponding parts will be equal, i.e. their sides, angles. Their area and perimeter will be the same too.
Two congruent triangles will definitely be equiangular, i.e. their corresponding angles will be the same.
However, vice-versa may not be true. That is, if two triangles are equiangular, then they need not be congruent. Two triangles having the same corresponding angles may still have sides of different length.
Property 2
If ∆ABC ≅ ∆PQR and ∆PQR ≅ ∆XYZ then ∆ABC ≅ ∆XYZ
Congruency rules of triangles
Now, let’s see the conditions which when fulfilled, will be said to be sufficient to prove congruency of the given triangles.
Two triangles will definitely be congruent to each other if they fulfil any one of the following criteria.
SSS Rule
This is the side-side-side congruence criterion.
If the corresponding sides of the two given triangles are equal in length, then it means that those two triangles are congruent.
AB = PQ; BC = QR; AC = PR
So, ∆ABC ≅ ∆PQR
SAS Rule
This is the side-angle-side congruence criterion.
Two triangles are congruent if:
- their two corresponding sides are equal in length, i.e. BC = QR; AC = PR, and
- the angle included by those two sides is also equal, i.e. ∠C = ∠R
So, ∆ABC ≅ ∆PQR
ASA Rule
This is the angle-side-angle congruence criterion.
Two triangles are congruent if:
- their two corresponding angles are equal, i.e. ∠A = ∠P; ∠B = ∠Q, and
- the side included by those two angles is also equal, i.e. AB = PQ
So, ∆ABC ≅ ∆PQR
AAS Rule
This is the angle-angle-side congruence criterion.
Two triangles are congruent if:
- their two corresponding angles are equal, i.e. ∠A = ∠P; ∠B = ∠Q, and
- one of the sides not included by those two angles is also equal, i.e. AC = PR, or BC = QR
So, ∆ABC ≅ ∆PQR
If we see ASA and AAS rules together, then we can say that two triangles will be congruent if:
- their two corresponding angles are equal, and
- any of their corresponding side is equal (whether that side is included within the two equal angles or not)
RHS Rule
In case of two right-angled triangles, they will be congruent if:
- their hypotenuses are equal, i.e. BC = QR
- one other side of one right-angled triangle is equal to corresponding side of the other right-angled triangle, i.e. AB = PQ, or AC = PR
So, ∆ABC ≅ ∆PQR
What is Similarity of Triangles?
Two triangles are said to be similar, if they are equal in shape, i.e. they have the same shape (but they need not be equal in size).
It is denoted by the symbol ~.
Notice the difference between Congruency and Similarity:
Congruent triangles are identical in every way - one of them can be made to superimpose on the other. They have the same shape and size.
Similar triangles have the same shape. Their size may differ, i.e. one triangle may be bigger/smaller than the other.
For example, if ∆ABC ~ PQR, then:
Corresponding angles are similar/equal: ∠A = ∠P; ∠B = ∠Q; ∠C = ∠R
Corresponding sides may or may not be equal.
Properties related to Similarity of triangles
Property 1
If two triangles are similar, then their corresponding sides will be proportional.
Similarly, all of their other corresponding parts (except angles) will be proportional too, e.g. medians, altitudes, angle bisectors, perpendicular bisectors etc.
Property 2
If two triangles are similar, then their area and perimeter will be proportional too.
Ratio between their perimeters = Ratio between their corresponding sides.
Ratio between their areas = Ratio between square of their corresponding sides.
Property 3
If ∆ABC ~ ∆PQR and ∆PQR ~ ∆XYZ then ∆ABC ~ ∆XYZ
Similarity rules of triangles
Now, let’s see the conditions which when fulfilled, will be said to be sufficient to prove similarity of the given triangles.
Two triangles will definitely be similar to each other if they fulfil any one of the following criteria.
AAA Rule
This is the angle-angle-angle similarity criterion.
If the corresponding angles of the two given triangles are equal, then it means that those two triangles are similar.
∠A = ∠P; ∠B = ∠Q; ∠C = ∠R
So, ∆ABC ~ ∆PQR
It is also called as AA rule, because if two corresponding angles of two triangles are equal, then it’s obvious that the third corresponding angle will be equal too.
For example, if ∠A = ∠P, and ∠B = ∠Q, then it’s certain that ∠C = ∠R.
SSS Rule
This is the side-side-side similarity criterion.
If the corresponding sides of the two given triangles are proportional, then it means that those two triangles are similar.
$\frac{AB}{PQ}$ = $\frac{BC}{QR}$ = $\frac{AC}{PR}$
So, ∆ABC ~ ∆PQR
SAS Rule
This is the side-angle-side similarity criterion.
Two triangles are similar if:
- their two corresponding sides are proportional, i.e. $\frac{BC}{QR}$ = $\frac{AC}{PR}$, and
- the angle included by those two sides is also equal, i.e. ∠C = ∠R
So, ∆ABC ~ ∆PQR